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IE 429, Parisay, January 2010

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- What you need to know from
- Probability and Statistics:
- Experiment outcome: constant, random variable
- Random variable: discrete, continuous
- Sampling: size, randomness, replication
- Data summary: mean, variance (standard deviation),
- median, mode
- Histogram: how to draw, effect of cell size
- Refer to handout on web page.

IE 429, Parisay, January 2010

- What you need to know from
- Probability and Statistics (cont):
- Probability distribution: how to draw, mass function,
- density function
- Relationship of histogram and probability distribution
- Cumulative probability function: discrete and continuous
- Standard distributions: parameters, other specifications
- Read Appendix C and D of your textbook.

IE 429, Parisay, January 2010

Relation betweenExponential distribution ↔ Poisson distribution

Xi : Continuous random variable, time between arrivals,

has Exponential distribution with mean = 1/4

X1=1/4 X2=1/2 X3=1/4 X4=1/8 X5=1/8 X6=1/2 X7=1/4 X8=1/4 X9=1/8 X10=1/8 X11=3/8 X12=1/8

0 1:00 2:00 3:00

Y1=3 Y2=4 Y3=5

Yi : Discrete random variable, number of arrivals per unit of time, has Poisson distribution with mean = 4. (rate=4) Y ~ Poisson (4)

IE 429, Parisay, January 2010

- What you need to know from
- Probability and Statistics (cont):
- Confidence level, significance level, confidence interval, half width
- Goodness-of-fit test
- Refer to handout on web page.

IE 429, Parisay, January 2010

Demo on Queuing Concepts

Refer to handout on web page.

Basic queuing system: Customers arrive to a bank, they

will wait if the teller is busy, then are served and leave.

Scenario 1: Constant interarrival time and service time

Scenario 2: Variable interarrival time and service time

Objective: To understand concept of average waiting time, average number in line, utilization, and the effect of variability.

IE 429, Parisay, January 2010

Scenario 1: Constant interarrival time (2 min) and

service time (1 min)

Scenario 2: Variable interarrival time and service time

Analysis of Basic Queuing System

Based on the field data

Refer to handout on web page.

T = study period

Lq = average number of customers in line

Wq = average waiting time in line

IE 429, Parisay, January 2010

Queuing Theory

Basic queuing system: Customers arrive to a bank, they

will wait if the teller is busy, then are served and leave.

Assume:

Interarrival times ~ exponential

Service times ~ exponential

E(service times) < E(interarrival times)

Then the model is represented as M/M/1

IE 429, Parisay, January 2010

Notations used for QUEUING SYSTEM in steady

state (AVERAGES)

= Arrival rate approaching the system

e = Arrival rate (effective) entering the system

= Maximum (possible) service rate

e = Practical (effective) service rate

L = Number of customers present in the system

Lq = Number of customers waiting in the line

Ls = Number of customers in service

W = Time a customer spends in the system

Wq = Time a customer spends in the line

Ws = Time a customer spends in service

IE 429

Analysis of Basic Queuing System

Based on the theoretical M/M/1

IE 429, Parisay, January 2010

Example 2: Packing Station with break and carts

Refer to handout on web page.

Objectives:

- Relationship of different goals to their simulation model
- Preparation of input information for model creation
- Input to and output from simulation software (Arena)
- Creation of summary tables based on statistical output for final analysis

IE 429, Parisay, January 2010

Example 2 Logical Model

IE 429, Parisay, January 2010

You should have some idea by now about the answer of these questions.

* What is a “queuing system”?

* Why is that important to study queuing system?

* Why do we have waiting lines?

* What are performance measures of a queuing system?

* How do we decide if a queuing system needs improvement?

* How do we decide on acceptable values for performance measures?

* When/why do we perform simulation study?

* What are the “input” to a simulation study?

* What are the “output” from a simulation study?

* How do we use output from a simulation study for practical applications?

* How should simulation model match the goal (problem statement) of study?

IE 429, Parisay, January 2010